deploy: c5d742b

Author: acbin

en/lc/319/index.html

@@ -13505,9 +13505,9 @@
       <ul class="md-nav__list">
 
           <li class="md-nav__item">
-  <a href="#solution-1" class="md-nav__link">
+  <a href="#solution-1-mathematics" class="md-nav__link">
     <span class="md-ellipsis">
-      Solution 1
+      Solution 1: Mathematics
     </span>
   </a>
 
@@ -75463,15 +75463,22 @@ <h2 id="description">Description</h2>
 </ul>
 
 <h2 id="solutions">Solutions</h2>
-<h3 id="solution-1">Solution 1</h3>
-<div class="tabbed-set tabbed-alternate" data-tabs="1:2"><input checked="checked" id="__tabbed_1_1" name="__tabbed_1" type="radio" /><input id="__tabbed_1_2" name="__tabbed_1" type="radio" /><div class="tabbed-labels"><label for="__tabbed_1_1">Python3</label><label for="__tabbed_1_2">Java</label></div>
+<h3 id="solution-1-mathematics">Solution 1: Mathematics</h3>
+<p>We can number the $n$ bulbs as $1, 2, 3, \cdots, n$. For the $i$-th bulb, it will be operated in the $d$-th round if and only if $d$ is a factor of $i$.</p>
+<p>For a number $i$, the number of its factors is finite. If the number of factors is odd, the final state is on; otherwise, it is off.</p>
+<p>Therefore, we only need to find the number of numbers from $1$ to $n$ with an odd number of factors.</p>
+<p>For a number $i$, if it has a factor $d$, then it must have a factor $i/d$. Therefore, numbers with an odd number of factors must be perfect squares.</p>
+<p>For example, the factors of the number $12$ are $1, 2, 3, 4, 6, 12$, and the number of factors is $6$, which is even. For the perfect square number $16$, the factors are $1, 2, 4, 8, 16$, and the number of factors is $5$, which is odd.</p>
+<p>Therefore, we only need to find how many perfect squares there are from $1$ to $n$, which is $\lfloor \sqrt{n} \rfloor$.</p>
+<p>The time complexity is $O(1)$, and the space complexity is $O(1)$.</p>
+<div class="tabbed-set tabbed-alternate" data-tabs="1:5"><input checked="checked" id="__tabbed_1_1" name="__tabbed_1" type="radio" /><input id="__tabbed_1_2" name="__tabbed_1" type="radio" /><input id="__tabbed_1_3" name="__tabbed_1" type="radio" /><input id="__tabbed_1_4" name="__tabbed_1" type="radio" /><input id="__tabbed_1_5" name="__tabbed_1" type="radio" /><div class="tabbed-labels"><label for="__tabbed_1_1">Python3</label><label for="__tabbed_1_2">Java</label><label for="__tabbed_1_3">C++</label><label for="__tabbed_1_4">Go</label><label for="__tabbed_1_5">TypeScript</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><table class="highlighttable"><tr><td class="linenos"><div class="linenodiv"><pre><span></span><span class="normal">1</span>
 <span class="normal">2</span>
 <span class="normal">3</span></pre></div></td><td class="code"><div><pre><span></span><code><span class="k">class</span> <span class="nc">Solution</span><span class="p">:</span>
     <span class="k">def</span> <span class="nf">bulbSwitch</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">n</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
-        <span class="k">return</span> <span class="nb">int</span><span class="p">(</span><span class="n">n</span> <span class="o">**</span> <span class="p">(</span><span class="mi">1</span> <span class="o">/</span> <span class="mi">2</span><span class="p">))</span>
+        <span class="k">return</span> <span class="nb">int</span><span class="p">(</span><span class="n">sqrt</span><span class="p">(</span><span class="n">n</span><span class="p">))</span>
 </code></pre></div></td></tr></table></div>
 </div>
 <div class="tabbed-block">
@@ -75486,6 +75493,36 @@ <h3 id="solution-1">Solution 1</h3>
 <span class="p">}</span>
 </code></pre></div></td></tr></table></div>
 </div>
+<div class="tabbed-block">
+<div class="highlight"><table class="highlighttable"><tr><td class="linenos"><div class="linenodiv"><pre><span></span><span class="normal">1</span>
+<span class="normal">2</span>
+<span class="normal">3</span>
+<span class="normal">4</span>
+<span class="normal">5</span>
+<span class="normal">6</span></pre></div></td><td class="code"><div><pre><span></span><code><span class="k">class</span><span class="w"> </span><span class="nc">Solution</span><span class="w"> </span><span class="p">{</span>
+<span class="k">public</span><span class="o">:</span>
+<span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">bulbSwitch</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
+<span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="n">sqrt</span><span class="p">(</span><span class="n">n</span><span class="p">);</span>
+<span class="w">    </span><span class="p">}</span>
+<span class="p">};</span>
+</code></pre></div></td></tr></table></div>
+</div>
+<div class="tabbed-block">
+<div class="highlight"><table class="highlighttable"><tr><td class="linenos"><div class="linenodiv"><pre><span></span><span class="normal">1</span>
+<span class="normal">2</span>
+<span class="normal">3</span></pre></div></td><td class="code"><div><pre><span></span><code><span class="kd">func</span><span class="w"> </span><span class="nx">bulbSwitch</span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
+<span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nb">int</span><span class="p">(</span><span class="nx">math</span><span class="p">.</span><span class="nx">Sqrt</span><span class="p">(</span><span class="nb">float64</span><span class="p">(</span><span class="nx">n</span><span class="p">)))</span>
+<span class="p">}</span>
+</code></pre></div></td></tr></table></div>
+</div>
+<div class="tabbed-block">
+<div class="highlight"><table class="highlighttable"><tr><td class="linenos"><div class="linenodiv"><pre><span></span><span class="normal">1</span>
+<span class="normal">2</span>
+<span class="normal">3</span></pre></div></td><td class="code"><div><pre><span></span><code><span class="kd">function</span><span class="w"> </span><span class="nx">bulbSwitch</span><span class="p">(</span><span class="nx">n</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
+<span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nb">Math</span><span class="p">.</span><span class="nx">floor</span><span class="p">(</span><span class="nb">Math</span><span class="p">.</span><span class="nx">sqrt</span><span class="p">(</span><span class="nx">n</span><span class="p">));</span>
+<span class="p">}</span>
+</code></pre></div></td></tr></table></div>
+</div>
 </div>
 </div>
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